Gödel Solution  

In General > s.a. relativistic cosmological models / causality; particle models; quantum field theory in curved spacetime.
* Idea: A set of rotating cosmological solutions of Einstein's equation with pressure-free perfect fluid matter on an \(\mathbb R\)4 manifold which shattered some preconceived ideas, since the local definition of inertial frame disagrees with the global one; It violates causality and is not time-orientable.
* Remark: A generalized form, actually described originally by Gödel, is a family of metrics with isometry group SO(2,1) × SO(2) × \(\mathbb R\).
* Properties: Each spacetime has a 5-parameter group of isometries which acts transitively, and there are closed timelike curves through all points; Any two points can be joined by a timelike curve; There are no non-empty achronal sets.
* Metric: Constructed from Maurer-Cartan forms of a suitable Lie group,

ds2 = –dt2 + dx2 – \(1\over2\)exp{2√2 ωx} dy2 + dz2 – 2 exp{√2 ωx} dt dy (partial) ,

with fluid 4-velocity u = ∂/∂t; ω = const = magnitude of vorticity of ua, 4πρ = ω2.

References
@ General: Gödel RMP(49) + GRG(00), in(52) + GRG(00); Kundt ZP(56); in Hawking & Ellis 73, 168–170; Bonnor et al CQG(98)gq/97 [exterior]; Németi et al a0811 [visualization]; Rindler in(09) + AJP(09)jun; Buser et al NJP(13)-a1303 [visualization]; Herrera et al PRD(13)-a1304 [physical meaning of the vorticity of the matter content]; Deszcz et al IJGMP(14) [curvature properties].
@ Energy and momentum: Dabrowski & Garecki CQG(02)gq/01, PRD(04)gq/03; Sharif IJMPA(03).
@ Closed timelike curves: Gleiser et al CQG(06)gq/05; Rosa & Letelier GRG(07)gq [stability]; Pfarr FP(10); Natário GRG(12)-a1105 [optimal time travel].
@ Light cones, causality: Pfarr GRG(81); Malament JMP(87); Ozsváth & Schucking AJP(03)aug; Dautcourt & Abdel-Megied CQG(06)gq/05.
@ Related topics: Saha & Chowdhury PS(00) [geodesic deviation]; Barrow & Tsagas CQG(04)gq/03 [stability]; Aydogdu & Salti gq/05-wd [spin-1 particle creation]; Sahdev et al gq/06 ['travel guide' and sketches]; Marecki gq/07-ch [wave equation]; Grave et al PRD(09) [geodesic motion, visualization]; Bartolo et al DG&A(11) [geodesic connectedness]; Pitanga a1201 [boundary, and the chronology protection conjecture]; > s.a. diffusion.

Similar Solutions and Other Theories > s.a. black holes in modified theories [5D Schwarzschild-Gödel]; brane world.
@ 2+1: Harriott & Williams GRG(01) [pfluid, with kink]; Sousa et al CQG(08)-a0705 [classification and properties]; Gürses GRG(10)-a0812; Brooks et al a1506 [classification, Cartan-Karlhede algorithm].
@ 5D: Rebouças & Teixeira JMP(98); Behrndt & Pössel PLB(04)ht/03 [supergravity, Λ < 0].
@ Variations: Romano & Goebel GRG(03) [with electromagnetic field]; Dryuma gq/05 [Riemann extension]; Dautcourt CQG(10)-a1009 [Gödel-like spacetimes, light cone]; Rendall JGP(11)-a1011 [topologically twisted]; > s.a. kerr-newman spacetime.
@ Chern-Simons-modified gravity: Konno et al PRD(08); Furtado et al PRD(09)-a0906, IJMPcs(12)-a1203; > s.a. higher-order theories.
@ Higher-order theories: Rebouças & Santos PRD(09)-a0906, Santos et al PRD(10)-a1004 [f(R) gravity]; Santos MPLA(13)-a1308 [f(R, T) gravity].
@ Other theories: Åman et al CQG(98)gq/97, Fonseca-Neto & Rebouças GRG(98)gq [Riemann-Cartan]; Barrow & Dabrowski PRD(98) [low-energy string theory]; Ozsváth & Schücking CQG(01) [newtonian analog]; Caldarelli & Klemm CQG(04)ht/03 [4D, supersymmetric]; Gürses et al CQG(05)ht/03, & Sarioglu CQG(05)ht [various dimensions]; Obukhov & Vargas PLA(04)gq [teleparallel gravity]; Gürses GRG(09)-a0801, & Şentürk GRG(16)-a1512 [Einstein-Æther theory]; Furtado et al PRD(11)-a1106 [in Hořava-Lifshitz gravity]; Furtado et al a1109 [in æther-modified gravity]; Ulhoa et al GRG(15)-a1503 [non-commutative corrections]; Agudelo et al PLB-a1603 [in Brans-Dicke theory].


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